Tuition Fees and University Enrollment:

A Meta-Regression Analysis ∗

Tomas Havraneka,b, Zuzana Irsovab, and Olesia Zeynalovab

aCzech National Bank

bCharles University, Prague

May 9, 2018

forthcoming in Oxford Bulletin in Economics and Statistics

Abstract

One of the most frequently examined relationships in education economics is the cor-

relation between tuition fee increases and the demand for higher education. We provide

a quantitative synthesis of 443 estimates of this effect reported in 43 studies. While large

negative estimates dominate the literature, we show that researchers report positive and

insignificant estimates less often than they should. After correcting for this publication

bias, we find that the literature is consistent with the mean tuition-enrollment elasticity

being close to zero. Nevertheless, we identify substantial heterogeneity among the reported

effects: for example, male students and students at private universities display larger elas-

ticities. The results are robust to controlling for model uncertainty using both Bayesian and

frequentist methods of model averaging.

Keywords: Enrollment, tuition fees, demand for higher education, meta-

analysis, publication bias, model averaging

JEL Codes: I23, I28, C52

∗Corresponding author: Zuzana Irsova, zuzana.irsova@ies-prague.org. Data and code are available inan online appendix at meta-analysis.cz/education. Havranek acknowledges support from the Czech ScienceFoundation grant #18-02513S, and Irsova acknowledges support from the Czech Science Foundation grant #16-00027S. This project has also received funding from the European Union’s Horizon 2020 Research and InnovationStaff Exchange program under the Marie Sklodowska-Curie grant agreement #681228 and Charles Universityproject PRIMUS/17/HUM/16. We thank the editor and four anonymous referees of the Oxford Bulletin ofEconomics and Statistics for useful comments. We are also grateful to Craig Gallet for providing us with thedataset used in his meta-analysis. The views expressed here are ours and not necessarily those of the CzechNational Bank.

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1 Introduction

The relationship between the demand for higher education and changes in tuition fees1 consti-

tutes a key parameter not only for deans but also for policymakers. It is therefore not surprising

that dozens of researchers have attempted to estimate this relationship. While the relationship

(often, but not always, presented in the form of an elasticity) can be expected to vary some-

what across different groups of students and types of universities, there has been no consensus

even on the mean effect, as many literature surveys demonstrate (see, for example, Jackson

& Weathersby, 1975; Leslie & Brinkman, 1987; Heller, 1997): the estimates often differ by an

order of magnitude, as we also show in Figure 1.

Figure 1: No clear message in 50 years of research

-1-.

50

.5

Effe

ct o

f tui

tion

on e

nrol

lmen

t (pa

rtial

cor

rel.

coef

ficie

nt)

1970 1980 1990 2000 2010 2020Publication year

Notes: The figure depicts a common metric (partial correlation co-efficient) of the reported effect of tuition fees on enrollment in highereducation institutions. The time trend is not statistically significant.

The academic discussion concerning the correlation between tuition fees and demand for

higher education dates back at least to Ostheimer (1953). Even though large price elasticities

do occasionally appear in the empirical literature (see, among others, Agarwal & Winkler, 1985;

Allen & Shen, 1999; Buss et al., 2004), the majority of the evidence corroborates the notion

of a rather price-inelastic demand for higher education across many contexts. Researchers

offer numerous explanations for the observed lack of large elasticities: for example, the effect

of financial aid compensating tuition changes (Canton & de Jong, 2002), increasing earnings

1For parsimony, in this paper, we usually omit “fee” and use the word “tuition” in its North American sense,“a sum of money charged for teaching by a college or university.”

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of graduates relative to those of non-graduates (Heller, 1997), historically small tuition fee

increases in real terms and the impact of aggressive marketing (Leslie & Brinkman, 1987),

larger student willingness to pay for quality (McDuff, 2007), expansion of the student pool with

female and minority participants, and the fact that many university students come from higher-

income families (Canton & de Jong, 2002). Even the very first literature review by Jackson

& Weathersby (1975) put forward the case for the correlation between tuition and enrollment,

while significant and negative, to be rather small in magnitude.

The existing narrative literature surveys, including Jackson & Weathersby (1975), McPher-

son (1978), Chisholm & Cohen (1982), Leslie & Brinkman (1987), and Heller (1997), place the

tuition-enrollment relationship below a 1.5 percentage-point change per $100 tuition increase.

The first quantitative review on this topic, Gallet (2007), puts the mean tuition elasticity of de-

mand for higher education at −0.6. However, every single review acknowledges that the mean

estimate could be somehow biased and driven by the vast differences in the design of stud-

ies, namely, methodological (Quigley & Rubinfeld, 1993), country-level (Elliott & Soo, 2013),

institution-level (Hight, 1975), and qualitative differences. Our goal in this paper is to exploit

the voluminous work of previous researchers on this topic, assign a pattern to the differences

in results, and derive a mean effect that could be used as “the best estimate for public policy

purposes” that the literature has sought to identify (Leslie & Brinkman, 1987, p. 189).

Achieving our two goals involves collecting the reported estimates of the effect of tuition

fees on enrollment and regressing them on the characteristics of students, universities, and

other aspects of the data and methods employed in the original studies. Such a meta-analysis

approach is complicated by two problems that have yet to be addressed in the literature on

tuition and enrollment: publication selection and model uncertainty. Publication selection

arises from the common preference of authors, editors, and referees for results that are intuitive

and statistically significant. In the context of the tuition-enrollment nexus, one might well

treat positive estimates with suspicion as few economists consider education to be Giffen good.

Nevertheless, sufficient imprecision in estimation can easily yield a positive estimate, just as it

can yield a very large negative estimate. The zero boundary provides a useful rule of thumb

for model specification, but the lack of symmetry in the selection rule will typically lead to an

exaggeration of the mean reported effect (Doucouliagos & Stanley, 2013).

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The second problem, model uncertainty, arises frequently in meta-analysis because many

factors may influence the reported coefficients. Nevertheless, absent clear guidance that would

specify which variables (out of the many dozen potentially useful ones) must be included in and

which must be excluded from the model, researchers face a dilemma between model parsimony

and potential omitted variable bias. The easiest solution is to employ stepwise regression, but

this approach is not appropriate because important variables can be excluded by accident in

sequential t-tests (this problem is inevitable, to some extent, also with more sophisticated meth-

ods of model selection—every time we need to choose which variables to exclude).2 In contrast,

we employ model averaging techniques that are commonly used in growth regressions: Bayesian

model averaging and frequentist model averaging, which are well described and compared by

Amini & Parmeter (2012). The essence of model averaging is to estimate (nearly) all models

with the possible combinations of explanatory variables and weight them by statistics related

to goodness of fit and parsimony.3

Our results suggest that the mean reported relation between tuition and enrollment is signif-

icantly downward biased because of publication selection (in other words, positive and insignif-

icant estimates of the relationship are discriminated against). After correcting for publication

selection, we find no evidence of a tuition-enrollment nexus on average. This result holds when

we construct a synthetic study with ideal parameters (such as a large dataset, control for endo-

geneity, etc.) and compute the implied “best-practice estimate”: this estimate is also close to

zero. Nevertheless, we find evidence of substantial and systematic heterogeneity in the reported

estimates. Most prominently, our results suggest that male students and students at private

universities display substantial tuition elasticities.

The paper is organized as follows. Section 2 describes our approach to data collection and

the basic properties of the dataset. Section 3 tests for the presence of publication selection bias.

Section 4 explores the data, method, and publication heterogeneity in the estimated effects of

tuition fees on enrollment and constructs a best practice estimate of the relationship. Section 5

provides extensions and robustness checks. Section 6 concludes the paper. An online appendix,

2Campos et al. (2005) provide a useful review of general-to-specific modeling.3Model averaging allows us to take into account the model uncertainty associated with our meta-analysis

model. Nevertheless, this approach does not address the model uncertainty in estimating the tuition-enrollmentnexus in primary studies: this second source of model uncertainty is the reason for conducting a meta-analysisin the first place (Stanley & Jarrell, 1989; Stanley & Doucouliagos, 2012). A technical treatment of these twosources of model uncertainty with relation to Bayesian model averaging is available in Appendix B of Havraneket al. (2017).

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available at meta-analysis.cz/education, provides the data and code that allow other researchers

to replicate our results.

2 The Dataset

Researchers often, but not always, estimate the tuition-enrollment relationship in the form of

the price elasticity of demand for higher education:

lnEnrollmentit = α+ PED · lnTuitionit + Y ED · ln Incomeit + Controlsijt + εit, (1)

where the demand for education Enrollment it typically denotes the total number of students

enrolled in higher education institution i in time period t, Tuition denotes the tuition pay-

ment for higher education, Income denotes the family income of a student, and its respective

coefficient YED denotes the income elasticity of demand. ε is the error term. The vector

Controlsijt represents a set of explanatory variables j, such as proxies for the quality of educa-

tion (university ranking, percentage of full professors employed, student/faculty ratio, average

score on assessment tests), funding opportunities (grants, external financial support, the cost

of loans), or labor market conditions (the level of unemployment or the wage gap between

university-educated and high school-educated workers).

From the empirical literature reporting the correlation between tuition fees and the demand

for higher education, we collect the coefficient PED. In (1), PED denotes the elasticity and

captures the percent change in demand for higher education if tuition increases by one percent.

The relationship between enrollment and tuition is, however, not always estimated in the lit-

erature in the form of an elasticity; sometimes other versions of (1) than log-log are used: the

relationship can be linear or represented by the student price response coefficient (Jackson &

Weathersby, 1975). Moreover, the definitions of the tuition and enrollment variables vary: while

tuition can represent net financial aid or include other fees, enrollment can represent the total

headcount of the enrolled, the number of applications, the percentage of enrolled students, or

enrollment probability. Even the uncertainty measure surrounding the point estimates reported

in the literature cannot always be converted into a standard error.

To be able to focus solely on elasticities and simultaneously make the sample fully com-

parable, we would need to eliminate a substantial part of the data (just as Gallet, 2007, did;

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moreover, our study faces an additional sample reduction since not all studies report an un-

certainty measure, which we need to account for in estimating publication bias). Maximizing

the number of observations and minimizing the mistakes made through conventional conversion

calls for a different type of common metric. McPherson (1978, p. 180) supports the case of

an ordinal measure: “There is probably not a single number in the whole enrollment demand

literature that should be taken seriously by itself. But a careful review of the literature will show

that there are some important qualitative findings and order-of-magnitude estimates on which

there is consensus, and which do deserve to be taken seriously.” Therefore, we use all estimates

of the tuition-enrollment nexus, including linear and semi-log specifications. We follow Doucou-

liagos (1995), Djankov & Murrell (2002), Doucouliagos & Laroche (2003), Babecky & Havranek

(2014), Valickova et al. (2015), and Havranek et al. (2016), among others, and convert the col-

lected estimates into partial correlation coefficients, which transform t-values to a measure that

is not related to the size of the dataset. Now, the PED coefficient is standardized to

PCC(PED)ij =T (PED)ij√

T (PED)2ij +DF (PED)ij

, (2)

where PCC(PED)ij represents the estimated partial correlation coefficient of the i-th estimate

of the tuition elasticity PED, with T (PED)ij representing the corresponding t-statistics and

DF (PED)ij representing the corresponding number of degrees of freedom reported in the j-th

study. We take advantage of the previously published surveys on this topic, especially Leslie &

Brinkman (1987), Heller (1997), and Gallet (2007), and extend the data sample by searching the

Google Scholar database. The search query is available online at meta-analysis.cz/education.

We added the last study on September 23, 2016.

The sample of studies we collect is subjected to three major selection criteria. First, the

study must investigate the relationship between tuition and enrollment with enrollment as the

dependent variable. This criterion eliminates multiple studies, including Mattila (1982), Galper

& Dunn (1969), and Christofides et al. (2001), which estimate only income effects on enrollment.

Second, the explanatory variable Tuition cannot be a dummy variable, which excludes studies

such as Bruckmeier & Wigger (2014) and Dwenger et al. (2012) (Hubner, 2012, for example,

uses a dummy variable indicating residence in a fee state to investigate the effects of tuition

on enrollment probabilities). Third, the study must report a measure of uncertainty around

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Table 1: Studies used in the meta-analysis

Agarwal & Winkler (1985) Doyle & Cicarelli (1980) Murphy & Trandel (1994)Alexander & Frey (1984) Elliott & Soo (2013) Noorbakhsh & Culp (2002)Allen & Shen (1999) Grubb (1988) Ordovensky (1995)Berger & Kostal (2002) Hemelt & Marcotte (2011) Parker & Summers (1993)Bezmen & Depken (1998) Hight (1975) Paulsen & Pogue (1988)Bruckmeier et al. (2013) Hoenack & Pierro (1990) Quigley & Rubinfeld (1993)Buss et al. (2004) Hoenack & Weiler (1975) Quinn & Price (1998)Campbell & Siegel (1967) Hsing & Chang (1996) Savoca (1990)Canton & de Jong (2002) Huijsman et al. (1986) Shin & Milton (2008)Chen (2016) Kane (2007) Suloc (1982)Cheslock (2001) King (1993) Tannen (1978)Chressanthis (1986) Knudsen & Servelle (1978) Toutkoushian & Hollis (1998)Coelli (2009) Koshal et al. (1976) Tuckman (1970)Craft et al. (2012) McPherson & Schapiro (1991)Dearden et al. (2011) Mueller & Rockerbie (2005)

the estimate (Corman & Davidson, 1984, for example, report neither t-statistics nor standard

errors). The final sample of studies used in our meta-analysis is listed in Table 1.

Previous literature surveys argue for a relatively modest magnitude of the relationship be-

tween tuition and enrollment (generally in terms of the mean student price response coefficient):

Jackson & Weathersby (1975), a survey of 7 studies published between 1967 and 1973, places the

enrollment change in the range of (−0.05,−1.46) percentage points per $100 tuition increase

in 1974 dollars; McPherson (1978) updates the range to (−0.05,−1.53). Leslie & Brinkman

(1987), a survey of 25 studies published between 1967 and 1982, places the mean student price

response coefficient at −0.7 per $100 in 1982 dollars; and Heller (1997), a survey of 8 studies

published between 1990 and 1996, reports a range of (−0.5,−1.0). The first literature survey

to examine quantitatively the heterogeneity in the estimates appears much later: Gallet (2007),

a meta-analysis of 295 observations from 53 studies published between 1953 and 2004, reports

a mean tuition elasticity of demand for higher education of −0.6.

Our final dataset covers 43 studies comprising 442 estimates of the relationship between

enrollment in a higher education institution and tuition recalculated to partial correlation co-

efficients. The oldest study was published in 1967, and the newest was published in 2016,

representing half of a century of research in the area. The (left-skewed) distribution of the

reported coefficients is shown in Figure 2; the coefficients range from −.941 to .707 and are

characterized by a mean of −0.171 and a median of −0.103. Approximately 25% of the es-

timates are larger than 0.33 in the absolute value, which, according to Doucouliagos (2011),

can be classified as a “large” partial correlation coefficient, while the mean coefficient is clas-

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Figure 2: Histogram of the partial correlation coefficients

025

5075

100

125

Freq

uenc

y

-1 -.5 0 .5 1Partial correlation coefficient

Notes: The figure depicts a histogram of the partial correlation co-efficients of the enrollment-tuition nexus estimates reported by indi-vidual studies. The dashed vertical line denotes the sample median,and the solid vertical line denotes the sample mean.

sified as a borderline “medium” effect. Nevertheless, using Cohen’s guidelines for correlations

in social sciences (Cohen, 1988), the mean effect of −0.171 would be classified as a “small”

effect. Although the histogram only has one peak, Figure 5 and Figure 6 (presented in the

Appendix) suggest substantial study- and country-level heterogeneity. Consequently, we collect

17 explanatory variables that describe the data and model characteristics and investigate the

possible reasons for heterogeneity below in Section 4.

Table 2 provides us with some preliminary information on the heterogeneity in the estimates.

Estimating the mean partial correlation via restricted maximum likelihood random effects meta-

analysis using the Hartung-Knapp modification (presented in the last row of the table) does

not affect our previous discussion regarding the average size of the effect in question. Next, we

summarize the simple mean values for each category and mean values weighted by the inverse

number of estimates reported per study (to assign each study the same weight) according

to different data, methodological, and publication characteristics. Larger studies with many

estimates largely drive the simple mean of the partial correlation coefficients, especially in

samples that consider private universities, female students, and countries outside the US (a

large portion of the estimates in the literature are estimated data from US universities, but

some studies focus on other countries, especially in Europe). Thus, it seems reasonable to focus

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on the weighted statistics in the following discussion. We observe differences between the short-

and long-term effects, which appear to be in line with intuition: a larger negative long-term

coefficient would suggest that in the long run, students have more time to search for other

competing providers of education. A substantial difference also appears when researchers do

not account for the presence of endogeneity in the demand equation: controlling for endogeneity

diminishes the partial correlation coefficient by 0.15; the effect itself is on the boundary between

a small and medium effect, according to Doucouliagos’s guidelines.

Table 2: Partial correlation coefficients for different subsets of data

Unweighted Weighted

No. of observations Mean 95% conf. int. Mean 95% conf. int.

Temporal dynamicsShort-run effect 209 -0.106 -0.134 -0.078 -0.135 -0.169 -0.101Long-run effect 233 -0.229 -0.269 -0.189 -0.233 -0.277 -0.189

Estimation techniqueControl for endogeneity 31 -0.034 -0.144 0.076 -0.043 -0.135 0.050No control for endogeneity 411 -0.181 -0.207 -0.155 -0.219 -0.249 -0.189

Data characteristicsPrivate universities 115 -0.086 -0.127 -0.044 -0.236 -0.284 -0.188Public universities 160 -0.198 -0.243 -0.153 -0.154 -0.207 -0.101Male candidates 49 -0.330 -0.389 -0.270 -0.329 -0.395 -0.263Female candidates 46 -0.252 -0.318 -0.186 -0.165 -0.220 -0.110

Spatial variationUSA 355 -0.150 -0.179 -0.121 -0.196 -0.229 -0.162Other countries 87 -0.256 -0.305 -0.207 -0.136 -0.171 -0.102

Publication statusPublished study 262 -0.249 -0.288 -0.211 -0.209 -0.247 -0.170Unpublished study 180 -0.056 -0.075 -0.038 -0.076 -0.106 -0.047

Publication yearUntil 1980 48 -0.227 -0.299 -0.155 -0.206 -0.308 -0.1031981–1990 80 -0.246 -0.342 -0.151 -0.129 -0.215 -0.0441991–2000 44 -0.191 -0.256 -0.127 -0.191 -0.259 -0.1232001–2010 144 -0.218 -0.254 -0.183 -0.208 -0.245 -0.170Since 2011 126 -0.040 -0.069 -0.011 -0.206 -0.263 -0.148

All estimates 442 -0.171 -0.196 -0.145 -0.186 -0.214 -0.158Random effects MA 442 -0.156 -0.180 -0.131 -0.167 -0.192 -0.142

Notes: The table reports mean values of the partial correlation coefficients for different subsets of data. The exact definitionsof the variables are available in Table 4. Weighted = estimates that are weighted by the inverse of the number of estimatesper study. MA = meta-analysis.

The evidence on one of the most widely studied topics in the literature, the difference in the

elasticity between public and private institutions, changes when weighting is applied: Hopkins

(1974), for example, finds that students in private institutions have a higher elasticity than

those in public universities, which is consistent with the weighted average from Table 2. The

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simple average is to some extent skewed by the considerable number of positive estimates in

larger studies (Grubb, 1988; Hemelt & Marcotte, 2011), which would correspond to a situation

in which private universities use tuition as a signal of the quality of the university. Male

candidates seem to display a larger elasticity to changes in tuition fees than female candidates,

and the difference increases when weighting is applied (the result is well in line with Huijsman

et al., 1986, but contradicts Bruckmeier et al., 2013, who do not find any differences). Spatial

differences do not seem to be extensive; however, we observe that published studies report larger

estimates of the effect of tuition on demand for higher education. Table 2 also shows that the

estimates do not vary much in time (the apparent drop in the estimates over the last decade

disappears when estimates are weighted). The differences in results between published and

unpublished studies might indicate the presence of publication bias, although not necessarily.

3 Publication Bias

Publication selection bias is especially likely to occur when there is a strong preference in the

literature for a certain type of result. Both editors and researchers often yearn for significant

estimates of a magnitude consistent with the commonly accepted theory. The law of demand,

which implies a negative relationship between the price and demanded quantity of a good,

is taken to be one of the most intuitive economic relationships; education is unlikely to be

perceived as a Giffen good (Doyle & Cicarelli, 1980).4 Therefore, researchers may treat positive

estimates of the tuition-enrollment nexus with suspicion and sometimes do explicitly refer to

the conventional expectation of the desired sign. (Canton & de Jong, 2002, p. 657), for example,

comment on their results as follows: “We find that the short-run coefficients all have the ‘right’

sign, except for the positive but insignificant coefficient on tuition fees . . . ”

Indeed, the unintuitive sign of an estimate might indicate identification problems; the proba-

bility of obtaining the ‘wrong’ sign increases with small samples, noisy data, or misspecification

of the demand function (Stanley, 2005). We should, however, obtain the unintuitive sign of

an estimate from time to time just by chance. Systematic under-reporting of estimates with

4On the other hand, it is important to mention the potential snob effect related to the price of educationand anticipated in the discussion of Table 2. While the association between prices and demand is unlikely to bepositive on average, it might easily be positive for some individual schools, students, or parents. This observationsuggests that in the case of higher education, positive price elasticities may be somewhat more acceptable than,for example, those in the literature on gasoline demand (Havranek et al., 2012).

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the ‘wrong’ sign drives the global mean in the opposite direction. This distortion of reported

results is a frequently reported phenomenon in economic research (for example, among other

studies, Doucouliagos & Stanley, 2013; Ioannidis et al., 2017; Havranek & Irsova, 2011, 2012;

Rusnak et al., 2013; Havranek & Kokes, 2015; Havranek et al., 2015b). Studies addressing the

law of demand are frequently affected by publication selection, but other areas also suffer from

bias, with the economics of education being no exception: Fleury & Gilles (2015) report pub-

lication bias in the literature on the inter-generational transmission of education, Ashenfelter

et al. (1999) find bias in the estimates of the rate of return to education, and Benos & Zotou

(2014) report bias toward a positive impact of education on growth. Primary studies could,

of course, incorporate theoretical expectations about the elasticity formally as priors within a

Bayesian estimation framework, but this approach is unfortunately not used in the literature

on the tuition-enrollment nexus.

The so-called funnel plot commonly serves as a visual test for publication bias (see, for

example, Stanley & Doucouliagos, 2010, and the studies cited therein). It is a scatter plot with

the effect’s magnitude on the horizontal axis and its precision (the inverse of the standard error)

on the vertical axis (Stanley, 2005). In the absence of publication bias, the graph resembles an

inverted funnel, with the most precise estimates close to the underlying effect; with decreasing

precision, the estimated coefficients become more dispersed and diverge from the underlying

effect. Moreover, if the coefficients truly estimate the underlying effect with some random

error, the inverted funnel should be symmetrical. The asymmetry in Figure 3 is consistent with

the presence of publication bias related to the sign of the effect; if the bias is related to statistical

significance, the funnel becomes hollow and wide. The literature exhibits a very similar pattern

of bias for the short- and long-term elasticity estimates; thus, in the calculations that follow,

we do not further divide the sample based on these two characteristics, but we control for the

differences in the next section.

Following Stanley (2005) and Stanley (2008), we examine the correlation between the partial

correlation coefficients PCCs and their standard errors in a more formal, quantitative way:

PCCij = PCC0 + β · SE(PCCij) + µij , (3)

where PCCij denotes i-th effect with the standard error SE(PCCij) estimated in the j-th study

and µij is the error term. The intercept of the equation, PCC0, is the ‘true’ underlying effect

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Figure 3: The funnel plot suggests publication selection bias

020

4060

80Pr

ecis

ion

of p

artia

l cor

rela

tion

coef

ficie

nt (1

/SE)

-1 1-.5 0 .5Effect of tuition on enrollment (partial correlation coefficient)

Long-runShort-run

Notes: The dashed vertical line indicates a zero partial correlationcoefficient of the elasticity of demand for higher education; the solidvertical line indicates the mean partial correlation coefficient. Whenthere is no publication selection bias, the estimates should be sym-metrically distributed around the mean effect.

absent publication bias; the coefficient of the standard error, β, represents publication bias. In

the case of zero publication bias (β = 0), the estimated effects should represent an underlying

effect that includes random error. Otherwise (β 6= 0), we should observe correlation between the

PCCs and their standard error, either because researchers discard positive estimates of PCCs

(β < 0) or because researchers compensate large standard errors with large estimates of PCCs.

In other words, the properties of the standard techniques used to estimate the tuition-enrollment

nexus yield a t-distribution of the ratio of point estimates to their standard errors, which means

that the estimates and standard errors should be statistically independent quantities.

Table 3 reports the results of (3). In Panel A, we present four different specifications applied

to the unweighted sample: simple OLS, an instrumental variable specification in which the

instrument for the standard error is the inverse of the square root of the number of observations

(as in, for example, Stanley, 2005; Havranek et al., 2018b); OLS, in which the standard error

is replaced by the aforementioned instrument (as in Havranek, 2015); and study-level between-

effect estimation.5 In Panel B, we weight all estimates by their precision, which assigns greater

importance to more precise results and directly corrects for heteroskedasticity (Stanley & Jarrell,

5It is worth noting that while we also intended to use study-level fixed effects (a common robustness checkaccounting for unobserved study-level characteristics, see the online appendix of Havranek & Irsova, 2017), thehigh unbalancedness of our panel dataset and the fact that a number of studies report only one observation makethis specification infeasible.

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Table 3: Funnel asymmetry tests detect publication selection bias

Panel A: Unweighted sample OLS IV Proxy Median

SE (publication bias) -1.142∗∗∗ -1.915∗∗∗ -1.523∗∗∗ -1.318∗∗

(0.36) (0.43) (0.27) (0.52)Constant (effect absent bias) -0.059 0.016 -0.004 -0.035

(0.06) (0.04) (0.04) (0.07)

Observations 442 442 442 442

Panel B: Weighted sample Precision Study

WLS IV OLS IV

SE (publication bias) -1.757∗∗∗ -2.305∗∗∗ -1.023∗∗∗ -1.753∗∗∗

(0.36) (0.49) (0.29) (0.52)Constant (effect absent bias) 0.001 0.026 -0.069∗∗ 0.015

(0.02) (0.02) (0.03) (0.05)

Notes: The table reports the results of the regression PCCij = PCC0 + β · SE(PCCij) + µij , where PCCij denotesi-th tuition elasticity of demand for higher education estimated in the j-th study and SE(PCCij) denotes its standarderror. Panel A reports results for the whole sample of estimates, and Panel B reports the results for the whole sample ofestimates weighted by precision or study. OLS = ordinary least squares. IV = the inverse of the square root of the numberof observations is used as an instrument for the standard error. Proxy = the inverse of the square root of the number ofobservations is used as a proxy for the standard error. Median = only median estimates of the tuition elasticities reportedin the studies are included. Study = model is weighted by the inverse of the number of estimates per study. Precision =model is weighted by the inverse of the standard error of an estimate. WLS = weighted least squares. Standard errors inparentheses are robust and clustered at the study and country level (two-way clustering follows Cameron et al., 2011). ∗

p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01.

1989), and furthermore, we weight all estimates by the inverse of the number of observations

per study, which treats small and large studies equally. In accordance with the mean statistics

from Table 2, the mean effect marginally increases (the mean relation between enrollment and

tuition changes becomes less sensitive) and even becomes significant but is still close to zero.

Two important findings can be distilled from Table 3. First, publication bias is indeed

present in our sample; according to the classification of Doucouliagos & Stanley (2013), the

magnitude of selectivity ranges from substantial (−2 > β > −1) to severe (β < −2). Sec-

ond, we cannot reject the hypothesis that the underlying tuition-enrollment effect corrected for

publication bias is zero. The estimated coefficient β suggests that the true effect is very small

or indeed zero. Nevertheless, Table 3 does not tell us whether data and method choices are

correlated with the magnitude of publication bias or the underlying effect. We address these

issues in the next section.

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4 Heterogeneity

4.1 Variables and Estimation

Thirty years ago, Leslie & Brinkman (1987) concluded their review of the tuition-enrollment

literature with disappointment regarding study heterogeneity: “Weinschrott (1977) was correct

when he warned about the difficulties in achieving consistency among such disparate studies.”

Data heterogeneity in our own sample is obvious from Figure 5 and Figure 6, presented in the

Appendix, and the substantial standard deviations of the mean statistics we report in Table 2.

Therefore, we code 17 characteristics of study design as explanatory variables that capture

additional variation in the data. The explanatory variables are listed in Table 4 and divided

into four groups: variables capturing methodological differences, differences in the design of

the demand function, differences in the dataset, and publication characteristics. Table 4 also

includes the definition of each variable, its simple mean, standard deviation, and the mean

weighted by the inverse of the number of observations extracted from a study.

Table 4: Description and summary statistics of regression variables

Variable Description Mean SD WM

Partial correlation coef. Partial correlation coefficient derived from the esti-mate of the tuition-enrollment relationship.

-0.171 0.271 -0.186

Standard error The estimated standard error of the tuition-enrollment estimate.

0.097 0.070 0.115

Estimation characteristicsShort-run effect = 1 if the estimated tuition-enrollment effect is

short-term (in differences) instead of long-term (inlevels).

0.473 0.500 0.480

OLS = 1 if OLS is used for the estimation of the tuition-enrollment relationship.

0.446 0.498 0.687

Control for endogeneity = 1 if the study controls for price endogeneity. 0.070 0.256 0.187

Design of the demand functionLinear function = 1 if the functional form of the demand equation

is linear.0.296 0.457 0.301

Double-log function = 1 if the functional form of the demand equationis log-log.

0.507 0.501 0.501

Unemployment control = 1 if the demand equation controls for the unem-ployment level.

0.495 0.501 0.350

Income control = 1 if the demand equation controls for income dif-ferences.

0.643 0.480 0.653

Data specificationsCross-sectional data = 1 if cross-sectional data are used for estimation

instead of time-series or panel data.0.204 0.403 0.303

Panel data = 1 if panel data are used for estimation instead ofcross-sectional or time-series data.

0.557 0.497 0.347

Continued on next page

14

Table 4: Description and summary statistics of regression variables (continued)

Variable Description Mean SD WM

Male candidates = 1 if the study estimates the tuition-enrollmentrelationship for male applicants only.

0.111 0.314 0.075

Female candidates = 1 if the study estimates the tuition-enrollmentrelationship for female applicants only.

0.104 0.306 0.051

Private universities = 1 if the study estimates the tuition-enrollmentrelationship for private universities only.

0.260 0.439 0.233

Public universities = 1 if the study estimates the tuition-enrollmentrelationship for public universities only.

0.362 0.481 0.454

USA = 1 if the tuition-enrollment relationship is esti-mated for the United States only.

0.803 0.398 0.839

Publication characteristicsPublication year Logarithm of the publication year of the study. 7.601 0.006 7.598Citations Logarithm of the number of citations the study re-

ceived in Google Scholar.3.529 1.079 3.227

Published study = 1 if the study is published in a peer-reviewed jour-nal.

0.593 0.492 0.828

Notes: SD = standard deviation, SE = standard error, WM = mean weighted by the inverse of the number ofestimates reported per study.

Estimation characteristics The exact distinction between short- and long-run effects is

disputable in most economic studies (see, for example Espey, 1998). If the author does not

clearly designate her estimate, we follow the basic intuition and classify the growth estimates as

short-term and the level estimates as long-term. Static models, however, introduce ambiguity.

If the dataset covers only a short period of time, the estimate might not reflect the full long-

term elasticity; thus, we label such estimates as short-run effects. Hoenack (1971) notes the

importance of temporal dynamics: lowering costs in the long run encourages students to apply

for higher education; in the short run, however, the change can only influence the current

applicants. The long-run effects are therefore likely to be larger. We do not divide the sample

between short- and long-run elasticities, which conforms to our previous discussion and the

practice applied by the previous meta-analyses on this topic (Gallet, 2007).

Researchers use various techniques to estimate the tuition-enrollment relationship. Fixed

effects in particular dominate the panel-data literature. More than one-third of the estimates

are a product of simple OLS, and surprisingly few studies control for endogeneity: as Coelli

(2009) emphasizes, an increase in tuition fees could be a response to an increase in the demand

for higher education. Therefore, the estimated coefficient may also include a positive price

response to the supply of student vacancies and thus underestimate the effect of tuition on the

demand for higher education (Savoca, 1990). For this reason, we could expect the estimates that

do not account for the endogeneity of tuition fees, such as those derived using OLS, to indicate

15

a correlation with enrollment different than those derived using, say, instrumental variables (as

in Neill, 2009, for example).6 To address this endogeneity bias, we include a dummy variable

indicating methods that do control for endogeneity.

Design of the demand function The relationship between tuition fees and the demand for

higher education can be captured in multiple ways. We present in (1) the double-log functional

form of the demand function, which produces the elasticity measure and accounts for half of

the estimates in our sample (Allen & Shen, 1999; Noorbakhsh & Culp, 2002; Buss et al., 2004).

Some authors, including McPherson & Schapiro (1991) and Bruckmeier et al. (2013), capture

the simple linear relationship between the variables using a linear demand function. Semi-

elasticities are also sometimes estimated and can be captured by the semi-log functional form

(Shin & Milton, 2008); several authors use non-linear Box-Cox transformations (such as Hsing

& Chang, 1996, who test whether the estimated elasticity is indeed constant). We suspect

that despite the transformation of all the estimates into partial correlation coefficients, some

systematic deviations in the estimates might remain based on the form of the demand function.

Researchers also specify demand equations to reflect various social and economic conditions

of the applicants. We account for whether researchers control for the two most important of

these conditions: the income level and unemployment rate. Lower-income students should be

more responsive to changes in tuition than higher-income students (McPherson & Schapiro,

1991); we expect systematic differences between results that do and do not account for income

differences. The effect of controlling for the unemployment level is not as straightforward. Some

authors (such as Berger & Kostal, 2002) hypothesize that the unemployment rate might be pos-

itively associated with enrollment, as attending a higher education institution can represent a

substitute for being employed. An unfavorable employment rate, by contrast, reduces the pos-

sibilities of financing higher education. Labor market conditions can also be captured by other

variables, for example, real wages, as the opportunity costs of attending university (Mueller &

Rockerbie, 2005) or a wage gap (Bruckmeier et al., 2013) reflecting the differences in earnings

between those who did and did not participate in higher education.

6Note that some studies, such as Coelli (2009), use OLS while simultaneously attempting to minimize endo-geneity bias using other than methodological treatments, mostly by including a detailed set of individual youthand parental characteristics.

16

Data specifications Leslie & Brinkman (1987) note that while cross-sectional studies reflect

the impact of explicit prices charged in the sample, panel studies reflect that each educational

institution implicitly accounts for the price changes of other institutions. Different projection

mechanisms could introduce heterogeneity in the estimates. Thus, we include a dummy variable

for studies that rely on cross-sectional variation and for studies that rely on panel data (the

reference category being time-series data). Since approximately 80% of our data are estimates

for the USA, we plan to examine whether geography induces systematic differences in the

estimated partial correlation coefficients. Elliott & Soo (2013) conduct a study of 26 different

countries including the US: the global demand for higher education seems to be more price

sensitive than US demand, although this conclusion is not completely robust.

The issue of male and female participants and their elasticities with respect to price changes

has also been discussed in previous studies. Savoca (1990) claims that females could face lower

earnings upon graduation; therefore, they may see higher education as a worse investment and

be less likely to apply. Bruckmeier et al. (2013) shows that gender matters when technical

universities are considered, while Mueller & Rockerbie (2005) find that male Canadian students

are more price sensitive than their female counterparts. McPherson & Schapiro (1991), however,

argue that the gender effect is in general constant across income groups, and Gallet (2007) does

not find significant gender-related differences in reported estimates.

The differences between public and private educational institutions are also frequently dis-

cussed, and researchers agree that these institutions face considerably different demand unless

student aid is provided. The results of Funk (1972) suggest the student price response to be

consistently lower for private universities. Hight (1975) supports these conclusions and argues

that the demand for community or public colleges tends to be more elastic than the demand

for private colleges. In a similar vein, Leslie & Brinkman (1987) note that the average student

at a private university has a higher family income base; furthermore, a lower-income student,

who is also more likely to enroll in a public university, typically demonstrates higher tuition

elasticities. However, Bezmen & Depken (1998) find those who apply to private universities to

be more price sensitive.

Publication characteristics While we do our best to control for the relevant data and

method features, it is unfeasible to codify every single difference among all estimates. There

17

might be unobserved aspects of data and methodology (or, more generally, quality) that drive

the results. For this reason, a number of modern meta-analyses (such as Havranek et al., 2015a)

employ a variable representing the publication year of the study: new studies are more likely

to present methodological innovations that we might have missed in our previous discussion.

Moreover, the equlibrium elasticity might have changed over time. It is plausible to argue that

earlier in the sample, higher education is more or less a luxury good. More recently, however,

with increasing higher education enrollment, higher education might have become more of a

necessity. Furthermore, we exploit the number of citations in Google Scholar to reflect how

heavily the study is used as a reference in the literature and information on publication status

since the peer-review process can be thought of as an indication of study quality.

The purpose of this section is to investigate which of the method choices systematically

influence the estimated partial correlation coefficients and whether the estimated coefficient

of publication bias from Section 3 survives the addition of these variables. Ideally, we would

like to regress the partial correlation coefficient on all 17 characteristics listed above, plus the

standard error. Since we have a relatively large number of explanatory variables, however,

it is highly probable that some of the variables will prove redundant. The traditional use of

model selection methods (such as eliminating insignificant variables one by one or choosing the

final model specification in advance) often leads to overly optimistic confidence intervals. In

this paper, we opt for model averaging techniques, which can address the model uncertainty

inherent in meta-analysis.

Bayesian model averaging (BMA) is our preferred choice of estimation technique to ana-

lyze heterogeneity. BMA processes hundreds of thousands of regressions consisting of different

subsets of the 18 explanatory variables. With such a large model space (218 models to be esti-

mated), we decide to follow some of the previous meta-analyses (such as Havranek & Rusnak,

2013; Irsova & Havranek, 2013; Havranek et al., 2018a, who also use the bms R package by

Feldkircher & Zeugner, 2009) and apply the Markov chain Monte Carlo algorithm, which con-

siders only the most important models. Bayesian averaging computes weighted averages of the

estimated coefficients (posterior means) across all the models using posterior model probabil-

ities (analogous to information criteria in frequentist econometrics) as weights. Thus, all the

coefficients have an approximately symmetrical distribution with a posterior standard deviation

18

(analogous to the standard error). Each coefficient is also assigned a posterior inclusion prob-

ability (analogous to statistical significance), which is a sum of posterior model probabilities

for the models in which the variable is included. Further details on BMA can be found, for

example, in Eicher et al. (2011).

When applying BMA, researchers have to make several choices. The first choice, as we

already mentioned, is whether to compute all models or to use the Markov chain Monte Carlo

approximation. Generally, with more than 15 variables, it becomes infeasible to compute all

models using a standard personal computer, so researchers typically approximate the whole

model space by using the model composition Markov chain Monte Carlo algorithm (Madigan

& York, 1995), which only traverses the most important part of the model space: that is, the

models with high posterior model probabilities. The second choice is the weight of the prior on

individual coefficients, the g-prior. The priors are almost always set at zero, which is considered

to be the safest choice, unless we have a very strong reason to believe that the coefficients

should have a particular magnitude (this is not the case in our study). The most commonly

used weight gives the prior the same importance as one individual observation: that is, very

little. This is called the unit information prior (UIP), and we apply it following Eicher et al.

(2011). The third choice concerns the prior on model probability. Again, the most commonly

used prior simply reflects that we have little knowledge ex ante, and so each model has the same

prior weight. Eicher et al. (2011) show that the combination of this uniform model prior and

the unit information g-prior performs well in predictive exercises. More technical details about

BMA in meta-analysis can be found in Appendix B of Havranek et al. (2017).

Although BMA is the most frequently used tool to address model uncertainty, recently pro-

posed statistical routines for frequentist model averaging (FMA) make the latter a competitive

alternative. Frequentist averaging, unlike the Bayesian version, does not require the use of ex-

plicit prior information. We follow Havranek et al. (2017), the first study to apply FMA in the

meta-analysis framework, who use the approach of Amini & Parmeter (2012), which is based on

on the works of Hansen (2007) and Magnus et al. (2010). As in the case of BMA, we attempt

to restrict our model space from the original 218 models and use Mallow’s model averaging

estimator (Hansen, 2007) with an orthogonalization of the covariate space according to Amini

& Parmeter (2012) to narrow the number of estimated models. Mallow’s criterion helps to

19

select asymptotically optimal weights for model averaging. Further details on this method can

be found in Amini & Parmeter (2012).

4.2 Results

The results of the BMA estimation are visualized in Figure 4. The rows in the figure represent

explanatory variables and are sorted according to the posterior inclusion probability from top

to bottom in descending order. The columns represent models and are sorted according to the

model inclusion probability from left to right in descending order. Each cell in the figure thus

represents a specific variable in a specific model; a blue cell (darker in grayscale) indicates that

the estimated coefficient of a variable is positive, a red cell (lighter in grayscale) indicates that

the estimated coefficient of a variable is negative, and a blank cell indicates that the variable is

not included in the model. Figure 4 also shows that nearly half of the variables are included in

the best model and that their signs are robustly consistent across different models.

A numerical representation of the BMA results can be found in Table 5 (our preferred spec-

ification is BMA estimated with the uniform model prior and unit information prior following

Eicher et al., 2011). Additionally, we provide two alternative specifications: first, a frequentist

check estimated by simple OLS with robust standard errors clustered at the study and country

level in which we include only variables from BMA with posterior inclusion probability higher

than 0.5. Second, we provide a robustness check based on FMA, which includes all explanatory

variables. All estimations are weighted using the inverse of the number of estimates reported

per study. In the next section, we also provide the robustness checks of BMA with different pri-

ors (following Fernandez et al., 2001; Ley & Steel, 2009) and different weighting (by precision).

Complete diagnostics of the BMA exercises can be found in Appendix B.

In interpreting the posterior inclusion probability, we follow Jeffreys (1961). The author

categorizes values between 0.5 and 0.75 as weak, values between 0.75 and 0.95 as positive,

values between 0.95 and 0.99 as strong, and values above 0.99 as decisive evidence for an effect.

Table 5 thus testifies to decisive evidence of an effect in the cases of Male candidates, Private

universities, and Citations; to positive evidence of an effect in the case of Panel data; and to

weak evidence of an effect in the cases of the Short-run effect, OLS, Income control, and USA

variables. While our robustness checks seem to support the conclusions from BMA, the evidence

20

Figure 4: Model inclusion in Bayesian model averaging

0 0.06 0.16 0.25 0.33 0.4 0.47 0.55 0.62 0.7 0.76 0.83 0.9 0.96

Male candidates

Citations

Private universities

Panel data

Standard error

OLS

Income control

USA

Short-run effect

Publication year

Endogeneity control

Published study

Unemployment control

Doublelog function

Cross-sectional data

Female candidates

Linear function

Public universities

Notes: The figure depicts the results of BMA. On the vertical axis, the explanatory variables are rankedaccording to their posterior inclusion probabilities from the highest at the top to the lowest at the bottom.The horizontal axis shows the values of the cumulative posterior model probability. Blue color (darker ingrayscale) = the estimated parameter of a corresponding explanatory variable is positive. Red color (lighterin grayscale) = the estimated parameter of a corresponding explanatory variable is negative. No color = thecorresponding explanatory variable is not included in the model. Numerical results are reported in Table 5.All variables are described in Table 4. The results are based on the specification weighted by the number ofestimates per study.

for an effect of the variables OLS and Control for endogeneity changes when FMA is employed.

Publication bias and estimation characteristics Although diminished to almost half of

its original value (Table 3), the evidence for publication bias represented by the coefficient on

the Standard error variable survives the inclusion of controls for data and method heterogeneity.

The result supports our original conclusion that publication bias indeed plagues the literature

estimating the relationship between tuition fees and the demand for higher education. The

evidence on the short-run effect is in line with expectation from Table 2 (and the conclusions of

21

Tab

le5:

Exp

lain

ing

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erog

enei

tyin

the

esti

mat

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men

tn

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s

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le:

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nm

odel

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ion

PC

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ost.

mea

nP

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SD

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Coef

.SE

p-v

alu

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alu

e

Con

stan

t0.

002

NA

1.0

00

-0.1

49

0.0

51

0.0

04

0.0

05

0.0

03

0.0

86

Sta

ndar

der

ror

-0.6

500.

439

0.7

58

-0.6

73

0.0

99

0.0

00

-0.7

12

0.2

52

0.0

05

Est

imati

on

chara

cter

isti

csShor

t-ru

neff

ect

0.05

20.0

53

0.575

0.0

93

0.0

07

0.0

00

0.1

38

0.0

38

0.0

00O

LS

-0.0

970.0

67

0.7

42

-0.1

05

0.0

44

0.0

18

-0.0

16

0.0

52

0.7

66

Con

trol

for

endog

enei

ty0.

052

0.0

73

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14

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65

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520.0

02

Des

ign

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the

dem

an

dfu

nct

ion

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ear

funct

ion

0.00

00.0

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0.0

64

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72

0.0

52

0.1

72

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ble

-log

funct

ion

-0.0

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150.0

95

-0.0

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34

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plo

ym

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0.0

26

0.1

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me

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083

0.0

650.7

18

0.0

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0.0

21

0.0

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91

0.0

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0.0

00

Data

spec

ifica

tion

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ross

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tion

aldat

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93

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Pan

eldat

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0.0

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0.7

84

-0.0

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Mal

eca

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ates

-0.3

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27

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81

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65

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Fem

ale

candid

ates

-0.0

070.0

40

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72

-0.1

45

0.1

10

0.189

Pri

vate

univ

ersi

ties

-0.1

690.0

39

0.9

96

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77

0.0

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00

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73

0.0

48

0.0

00

Public

univ

ersi

ties

0.00

10.0

12

0.060

0.0

11

0.0

38

0.7

67

USA

-0.0

950.0

86

0.6

43

-0.0

38

0.0

36

0.2

83

-0.1

96

0.0

60

0.0

01

Pu

blic

ati

on

chara

cter

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tion

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130.0

18

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16

0.0

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12

Cit

atio

ns

0.04

30.0

10

1.0

00

0.0

33

0.0

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0.0

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53

0.0

10

0.0

00

Publish

edst

udy

-0.0

150.0

37

0.1

83

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54

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52

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01

Stu

die

s43

43

43

Obse

rvat

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442

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442

No

tes:

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22

Gallet, 2007): its positive coefficient suggests a lower sensitivity to price changes in the short-

run than in the long-run, when the enrollees have more time to adapt to a new pricing scheme

and search for adequate substitutes.

Table 5 reports that the evidence on the importance of the OLS and Control for endogeneity

variables is mixed across different model averaging approaches. The instability of the two

coefficients is somewhat intuitive: studies using simple OLS rarely control for endogeneity; the

correlation coefficient of these variables is −0.45. The direction of the effect of controlling for

endogeneity that we identify is, however, not consistent with what is often found in the literature

(Savoca, 1990; Neill, 2009): estimates that do not account for endogeneity are expected to show

smaller effects since these estimates may capture the positive effects of price on the supply of

education. Our results suggest that controlling for endogeneity understates the reported effects

(although the corresponding posterior inclusion probability is less than 0.5, suggesting a very

weak link). This finding is consistent with Gallet (2007), who reports that methods controlling

for endogeneity generate more positive estimates than does OLS, and it suggests a potential

problem with approaches to endogeneity control employed in the literature. Alternatively, the

finding might also imply that economists do not fully understand the demand for education.

Several techniques are commonly used that attempt to purge the effects of endogeneity from

the demand equation, but difficulties for researchers often arise while doing so. In the presence

of endogeneity, OLS should directly lead to biased and inconsistent estimates, although some

researchers justify its utilization by the identification of the supply and demand side. First,

with public financing and/or constant operating costs supported by revenues collected from

new enrollees, the universities may supply more enrollments without raising tuition (King,

1993). When applicants outstrip enrollment (which they often do), the tuition price in the face

of excess demand does not clear the market for higher education. Researchers then assume the

supply of admission at the market level of tuition to be infinitely elastic and all independent

variables to explain only the demand side (justifying the utilization of OLS as in Mueller &

Rockerbie, 2005). Second, as Coelli (2009) states, one could explain the demand side in such

a detailed manner that would obliterate the remaining endogeneity. For this purpose, Coelli

(2009) uses a set of rich individual student and parental characteristics.

The perfect elasticity of the supply side is, however, rather academic (we can never rule out

23

the correlation between the explanatory variable and the error term of the demand equation

with certainty). Moreover, detailed micro-level information usually comes from surveys and

is unavailable for most researchers, which makes the simultaneous equation model difficult to

estimate. The literature treating endogeneity thus relies on instrumental variables instead. In-

strumental variables are defined as tuition correlates that should not directly affect the demand

for higher education. Choosing the appropriate instrument might be tricky: the instrument

must be strong enough to provide a source of variation for the model but must still show an

exogenous source of variation. Also the fixed-effect instrumental variable estimation with a

weak instrument can easily lead to results that are no better than those of OLS.

The lagged level of the endogenous variable represents a popular choice for an instrument

(Allen & Shen, 1999). The tuition fee is typically further instrumented by an income level

and unemployment rate (sometimes in lags, which reflect more realistic delayed adjustments

in tuition), determinants of education costs (excess-tax on tuition, faculty salaries), and time

dummies capturing other possible external influences. Two-stage and three-stage least squares

typically appear in applications (Berger & Kostal, 2002; Savoca, 1990). The generalized-method-

of-moments estimator (Arellano & Bond, 1991) is another frequently chosen technique. Never-

theless, contrary to the recommendations of Roodman (2009), many studies do not report the

number of instruments used in the analysis. In a nutshell, we believe that even the results of the

studies that claim to successfully control for endogeneity cannot be automatically interpreted

causally.

Design of the demand function According to our results, the functional form of the demand

function does not systematically affect the reported coefficients. This conclusion differs from the

findings of Gallet (2007), who argues that the outputs of semi-log, linear, and Box-Cox functional

forms are significantly different from the results produced by directly estimating the double-log

demand function. Furthermore, the inclusion of the control variable for unemployment also

does not seem to drive the estimated sensitivity of enrollment to tuition changes; the control

for an individual’s income group, however, significantly decreases the estimated sensitivity.

Data specifications Leslie & Brinkman (1987) report that estimates produced from cross-

sectional datasets and time-series datasets do not vary substantially, and our results support this

24

conclusion. Panel data, which combine both cross-sectional and time information, however, lead

to partial correlation coefficients that are 0.11 smaller, other things being equal. We also argue

that male students exhibit a systematically larger (by 0.35) sensitivity to changes in tuition in

comparison with the general population, which is in contrast to the results of Gallet (2007), who

finds that gender-related characteristics fail to significantly affect the reported tuition elasticity.

The results are, however, in line with those of Mueller & Rockerbie (2005), who find males to

be more price sensitive than females. As an explanation, Mueller & Rockerbie (2005) argue

that since the rate of return to a university degree might be higher for a female than for a male,

females are willing to spend more on tuition fees.

Some studies estimate the effect of tuition in public universities, while others consider private

universities. We find that candidates applying to private universities display larger tuition

elasticities. One interpretation of the different magnitudes of the price sensitivity is that the

more or the better the substitutes are for a particular commodity, the higher the price sensitivity.

In our case, students should be able to switch more easily to a substitute institution when private

university tuition rises than when public university tuition rises, as the pool of substitutes for

private institutions should be larger and also includes public universities (where costs are lower).

These results, however, contradict those of Leslie & Brinkman (1987) and Hight (1975), who note

that the average enrollee at a private university is rich and, thus, less price-elastic. Estimates for

the US seem to be less negative than those for other countries. We would argue that given the

extent of the US system of higher education, the pool of close substitutes might be larger in the

US than in the rest of the world, where a single country hosts a smaller number of universities.

Publication characteristics There are two results on publication characteristics that are

consistent with the meta-analysis of Gallet (2007): the insignificance of publication year and

publication status. Publication year may capture changes in methodological approaches; never-

theless, Table 5 indicates that the newer studies do not report systematically different results.

Further, we show in Table 2 that the partial correlation coefficients reported in Published stud-

ies are arguably smaller than those in unpublished or unrefereed studies. The impact of other

explanatory variables, however, erases this link; in fact, Table 5 suggests the publication status

of a study does not matter for the magnitude of the estimates. More important is how much

25

attention the paper attracts from readers, which is captured by the number of citations. Highly

cited articles report less-sensitive estimates of the tuition-enrollment relationship.

Thus far, we have argued that the mean reported value of the tuition-enrollment partial

correlation coefficient, −0.19 (shown in Table 2), is significantly exaggerated by the presence

of publication bias. The effect absent publication bias, shown in Table 3, is close to zero.

We have also seen that the effect is substantially influenced by data, method, and publication

characteristics. To provide the reader with a ‘rule-of-thumb’ mean effect that controls for

all these influences and potential biases, we construct a synthetic ‘best-practice’ study that

employs our preferred choices with respect to all the sources of heterogeneity in the literature.

The definition of best practice is subjective, but it is a useful check of the combined effect of

various misspecifications and publication bias. Essentially, we create a weighted average of all

estimates by estimating fitted values from the BMA and FMA specifications.

The ideal study that we imagine would be published in a refereed journal, highly cited, and

recent; thus, we set all publication characteristics at the sample maxima (we censor, however,

the number of citations at the 99% level due to the presence of outliers—although using the

sample maximum would provide us with an even stronger result). We remove any sources

of publication and endogeneity bias; thus, we set the standard error and OLS at the sample

minima and the control for endogeneity at the sample maximum. We prefer the usage of broader

datasets and favor the inclusion of controls for the economic environment, and thus, we set the

panel dataset and controls for income and unemployment at the sample maxima. Moreover, we

prefer the double-log functional form since it directly produces an elasticity and represents a

measure with a clear interpretation that is independent of the current price level. We leave the

remaining variables at their sample means.

The ‘best-practice’ estimation in Table 6 yields a partial coefficient of −0.037 with a 95%

confidence interval of (−0.055; −0.019). The estimated standard errors are relatively small,

and even with plausible changes to the definition of best practice (such as changing the design

of the demand function), the results reported in Table 6 change only at the third decimal

place. The best-practice estimation thus corroborates our previous assertions regarding the

correlation between tuition and enrollment: in general, we observe higher price elasticities in

26

Table 6: Best practice estimation yields a tuition-enrollment effect that is close to zero

Bayesian model averaging Frequentist model averaging

Mean 95% conf. int. Mean 95% conf. int.

Short-run effect -0.010 -0.032 0.012 0.069 0.047 0.090Long-run effect -0.062 -0.081 -0.042 -0.070 -0.089 -0.050Private universities -0.167 -0.190 -0.144 -0.141 -0.164 -0.117Public universities 0.003 -0.033 0.039 0.043 0.007 0.079Male candidates -0.361 -0.529 -0.192 -0.348 -0.517 -0.180Female candidates -0.017 -0.120 0.087 -0.112 -0.216 -0.008

All estimates -0.037 -0.055 -0.019 -0.003 -0.021 0.015

Notes: The table presents mean estimates of the partial correlation coefficients implied by theBayesian/frequentist model averaging and our definition of ‘best practice.’ Because BMA does not workwith the concept of standard errors, the confidence intervals for BMA are approximate and constructed usingthe standard errors estimated by simple OLS with robust standard errors clustered at the study and countrylevel.

the long-run, higher elasticities among individuals enrolled in private universities, and higher

elasticities among male students. The overall mean, however, is very close to zero.

5 Extensions and Robustness Checks

In this section, we pursue seven modifications of our baseline BMA model presented earlier.

The results of these modifications are divided into two tables, one entitled ‘robustness checks’

(four specifications) and the other entitled ‘extensions’ (three specifications). Roughly speaking,

robustness checks show the sensitivity of our main results to plausible changes in estimation

strategy, while extensions provide new results. We start by discussing Table 7, which contains

the robustness checks. The structure of the table is similar to what we have already seen in the

previous section: the results shown include the posterior mean, posterior standard deviation,

and posterior inclusion probability for each variable. The table is divided into four vertical

panels, which mark different specifications.

The first specification in Table 7 uses a set of priors different from the one in our baseline

model. In this robustness check, we follow Fernandez et al. (2001) and choose BRIC for g-prior;

for model size, we use the beta-binomial random prior advocated by Ley & Steel (2009). Along

with the plain vanilla UIP g-prior and uniform model prior used in the baseline model, the

choice of priors in the robustness check constitutes arguably the most common combination of

priors used in BMA. In our experience, the choice between UIP and BRIC rarely is material for

BMA results. In contrast, the choice of model prior can have important effects.

27

The uniform model prior relies on the intuitive notion that each model, irrespective of the

number of variables included, should have the same weight in results. Among all the possible

models, the most common are those that contain the mean number of these variables (and

the least common are those that include only a couple of them or almost all of them). In

consequence, the uniform model prior puts more weight on models of mean size. An appealing

alternative is to place the prior directly on model size, which is what we do here: we assume

that ex ante, each model size has the same probability. In this application, the effect of this

change on results is minimal. The estimated coefficients change little; we still obtain evidence of

significant publication bias, and our discussion of the impact of individual variables would not

change at all. This is an encouraging finding of robustness to a choice in priors, strengthened

further by the fact that frequentist model averaging (which relies on a completely different

econometric philosophy), presented in the previous section, delivers similar results as well.

In the second panel of Table 7, we use weights proportional to the reported precision of the

estimates. These are appealing weights because they address the inherent heteroskedasticity

problem in meta-analysis, and intuitively, it makes sense to give more weight to more precise

results. Indeed, throughout the analysis, we have treated the estimates of the tuition-enrollment

nexus as data points, although the estimates have uncertainty attached to them. The appendix

to Havranek et al. (2017) shows technically how this uncertainty affects the results of Bayesian

model averaging. In sum, our posterior inclusion probabilities might be exaggerated. This

problem can be addressed in two ways: first, by estimating OLS or frequentist model averaging

with robust standard errors (as we did in the previous section), and second, by estimating

a BMA model with weights proportional to precision. This is an intuitive solution that also

follows the literature on estimated dependent variable models, but such a precision-weighted

BMA exercise lacks rigorous Bayesian foundations, as is also discussed by Havranek et al. (2017).

Nevertheless, we proceed with the robustness check.

The precision-weighted BMA is more difficult to interpret because of increased collinearity.

This issue is especially serious for variables that show little variation within studies; by using

weights proportional to precision, we generate artificial correlation between these variables. In

consequence, it is hard to comment on individual variables—with the exception of the standard

error itself. We see from Table 7 that while the corresponding coefficient retains its negative sign,

its magnitude drops. Additionally, the posterior inclusion probability decreases, although it still

28

Tab

le7:

Exp

lain

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het

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the

esti

mat

es(r

obu

stn

ess

chec

ks

ofT

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Res

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Diff

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tB

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pri

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Pre

cisi

on

-wei

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dd

ata

Ela

stic

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pati

alva

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ion

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nP

CC

Post

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ean

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DP

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ean

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t.S

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ost

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ean

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DP

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ost.

mea

nP

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DP

IP

Con

stan

t0.

002

NA

1.000

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17

NA

1.000

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11

NA

1.0

000.0

02

NA

1.000

Sta

nd

ard

erro

r-0

.614

0.44

40.7

28

-0.2

77

3.915

0.5

13-0

.860

0.2

26

0.9

96

-0.6

60

0.4

390.7

64

Est

imati

on

chara

cter

isti

csS

hor

t-ru

neff

ect

0.05

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540.6

17

0.023

0.0

31

0.4

17

0.3

74

0.2

09

0.8

79

0.051

0.0

53

0.5

62

OL

S-0

.096

0.067

0.7

42-0

.001

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75

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58

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97

0.0

660.7

47

Con

trol

for

end

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eity

0.05

40.0

73

0.417

0.0

12

0.0

24

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62

0.256

0.2

56

0.5

77

0.0

51

0.0

72

0.4

06

Des

ign

of

the

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an

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ear

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38

1.0

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01

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11

0.0

62D

ou

ble

log

fun

ctio

n-0

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0.0

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01

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54

0.0

54

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39

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15

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94

Un

emp

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ent

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560.0

74

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me

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13

Data

spec

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ross

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tion

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160.1

98

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65

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51

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90

Pan

eld

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46

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57

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00

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12

0.0

72

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Male

can

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vate

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bli

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ties

0.00

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80.5

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ns

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le4.

29

surpasses the 0.5 threshold: the underlying model is likely to include a variable that corrects for

publication bias. Perhaps the most useful comparison between this robustness check and our

baseline model involves computing the best-practice estimate: after all, that is the main output

of our analysis. Because such an estimate uses the specification as a whole, it is not affected

by collinearity problems. Here, we get a best-practice estimate of −0.04, which is very close to

our baseline result and consistent with the notion that after correcting for publication bias and

misspecifications, there is little evidence for any strong tuition-enrollment nexus.

The third panel of Table 7 focuses on the subset of estimates that are computed using the

log-log specification. Consequentially, they can be interpreted as elasticities and allow for more

direct discussion of the underlying economic effect of tuition fees. This is obviously a useful

robustness check, but we have to exclude approximately 50% of the dataset for which we are

unable to reconstruct elasticities (very often, sample means for the data are not reported in

the literature, so we cannot approximate elasticities for linear and other estimates). It is also

clear that we have to omit from the BMA exercise variables that reflect the functional form

used in primary studies, because now we only focus on log-log specifications. The estimated

model thus changes somewhat, but evidence for publication bias is even stronger than in our

baseline model: the posterior inclusion probability for this variable reaches 0.996. The implied

best-practice estimate is, once again, statistically indistinguishable from zero.

Table 8: Partial correlation coefficients for different subsets of data (additional variables)

Unweighted Weighted

No. of observations Mean 95% conf. int. Mean 95% conf. int.

HeterogeneityIndividual level 85 -0.294 -0.352 -0.236 -0.234 -0.299 -0.168University level 357 -0.141 -0.169 -0.114 -0.174 -0.206 -0.143Undergraduate enrollees 288 -0.118 -0.148 -0.088 -0.169 -0.206 -0.133Graduate enrollees 34 -0.420 -0.523 -0.316 -0.342 -0.421 -0.264Freshman enrollees 132 -0.124 -0.163 -0.084 -0.190 -0.243 -0.138MBA program 8 -0.324 -0.462 -0.186 -0.328 -0.458 -0.199

Spatial variationCanada 51 -0.364 -0.430 -0.299 -0.200 -0.270 -0.130Netherlands 24 -0.093 -0.139 -0.046 -0.063 -0.105 -0.022

All estimates 442 -0.171 -0.196 -0.145 -0.186 -0.214 -0.158

Notes: The table reports mean values of the partial correlation coefficients for different subsets of data. The exact definitionsof the variables are available in Table 4. Weighted = estimates that are weighted by the inverse of the number of estimatesper study.

In the last panel of Table 7, we add two more dummy variables reflecting spatial variation in

30

the estimates of the tuition-enrollment nexus. A large majority of the estimates in our sample

was derived from US data, but the next two countries with the largest number of estimates are

Canada and the Netherlands (in total, our sample covers 21 countries, but for many of them,

we only have a few estimates, so it is unfeasible to include dummy variables for all of them).

Controlling for potential systematic differences in estimates for these two countries does not

change our results, and the two dummy variables have low posterior inclusion probabilities (less

than 0.1).

Now, we turn our attention to extensions of the baseline model. Sometimes it is not easy

to draw a line between robustness checks and extensions (for example, the additional country

dummies in the last panel of Table 7 might also be considered an extension), but the distinction

is useful for presentation purposes. The core of the extensions presented in Table 9 consists

of the addition of new potentially useful variables into the model. The reason that we do not

consider these variables in the baseline model is the same reason that we do not, in the baseline,

use weights that are proportional to precision: the inclusion of these new variables greatly

increases collinearity, so we find it more useful to present them in a separate exercise. The new

variables are first presented in Table 8, where we show sample means for the respective groups

of estimates.

First, we include a dummy variable Individual level, which equals one for estimates that

are computed using student-level data, as opposed to more aggregated data available at the

university level. Individual-level estimates are relatively rare in the literature and, on average,

seem to show larger partial correlation coefficients for the tuition-enrollment nexus. Next, we

account for the type of study program under examination: undergraduate, graduate, and MBA.

We also include an additional dummy variable that equals one if the study focuses on freshman

students. We observe larger partial correlation coefficients for graduates and MBAs (which

makes good sense, because these products still have a special status, and students can easily

choose to work instead, thereby increasing the price elasticity), but it is important to note that

for these groups, we have relatively few observations.

In the first panel of Table 9, we explore the consequences of adding these new variables into

our baseline model. In addition to variables listed in Table 8, we also include the square root of

the number of observations used in the primary study: because partial correlation coefficients

31

Tab

le9:

Exp

lain

ing

het

erog

enei

tyin

the

esti

mat

es(e

xte

nsi

ons

ofT

able

5)

Res

pon

seva

riab

le:

Ad

dit

ion

al

vari

ab

les

Tim

eva

riati

on

Com

bin

ed

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itio

nP

CC

Pos

t.m

ean

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.S

DP

IPP

ost

.m

ean

Post

.SD

PIP

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ean

Post

.S

DP

IP

Con

stan

t0.

001

NA

1.0

00

-0.0

01

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1.0

00

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04

NA

1.0

00

Sta

nd

ard

erro

r-0

.160

0.2

98

0.2

77

-1.0

89

0.2

94

0.9

89

-1.3

92

0.2

48

0.9

99

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imati

on

chara

cter

isti

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hor

t-ru

neff

ect

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69

0.0

45

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93

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01

0.0

09

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OL

S-0

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75

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61

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69

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trol

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eity

0.06

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0.6

13

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ble

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ion

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ent

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62

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me

contr

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0.0

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Data

spec

ifica

tion

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ross

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tion

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ata

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32

are a function of the number of observations, this variable may be important by definition.

As we have noted earlier, collinearity increases above acceptable limits (many variables now

have variance-inflation factors above 10), so the interpretation of the signs and magnitudes of

individual coefficients should be taken with a grain of salt. That being said, the parameter

corresponding to the magnitude of publication bias retains its negative sign but loses much of

its importance (the posterior mean is now only −0.16 and posterior model probability is 0.28).

Concerning the additional variables, two findings arise that also survive a later robustness check:

first, ceteris paribus, studies using student-level data tend to find larger partial correlation

coefficients than studies using university level data; second, MBA programs are also associated

with a stronger tuition-enrollment nexus. Undergraduate and graduate study programs seem

to display similar price elasticities, and the correlation between the number of observations and

reported partial correlation is small (and disappears in a robustness check).

The second panel of Table 9 shows how the effect of selected variables changes in time. For

this extension we choose five important variables: standard error (reflecting publication bias),

the use of OLS, control for endogeneity, use of cross-sectional data, and use of panel data. In

our baseline BMA model, we find no systematic influence of the publication year of studies on

the reported results. In contrast, we find evidence of systematic effects related to the use of

OLS and panel data. In this extension, we find no evidence that the magnitude of publication

bias would change in time (the interaction of the reported standard error and the publication

year of the study has a posterior inclusion probability of about 0.12). A large posterior inclusion

probability (1) is reported for the interaction between the use of panel data and publication

year. The interaction is negative, which means that the negative effect of the use of panel data

on the reported results identified in our baseline model has been strengthening in recent years.

In the third column of the table, we merge both extensions, which provides a simple robustness

check. Overall, these two extensions suggest that data aggregation systematically affects results

and that there is limited evidence for the literature converging to a consensus (which could be

reflected, for example, by a trend in publication bias).

33

6 Concluding Remarks

In this paper, we conduct a quantitative synthesis of 442 estimates of the relationship between

tuition fees and the demand for higher education reported in 43 studies. Our contribution on top

of the previous meta-analysis by Gallet (2007) is twofold: first, we include a formal treatment of

publication bias, and second, we include a treatment of model uncertainty using model averaging

methods when searching for the determinants of the underlying effect. The literature shows

substantial publication selection against positive estimates, suggesting that many researchers use

the sign of the estimated effect as a specification test (education is unlikely to be a Giffen good).

The mean effect beyond publication bias is close to zero. When we attribute greater weight

to the more reliable estimates (published in respected journals and derived using appropriate

methodology), we obtain a similarly small mean estimate of the tuition-enrollment nexus.

We also find evidence for systematic dependencies between the estimated effects and data,

methodological, and publication characteristics. Male students display larger tuition elasticities,

as do students at private universities. Previous research has yielded mixed results on both of

these relationships. Our findings concerning male students are consistent with those of Mueller

& Rockerbie (2005), who argue that because female students tend to have a higher rate of return

from university education, they are willing to spend more on tuition fees. Concerning private

universities, it might be easier for their students to find substitutes if tuition increases; for

public university students, a large portion of the market (most private universities) is already

unaffordable. Next, we find that highly cited studies tend to report little correlation between

enrollment and tuition, although the direction of causality is unclear. Our results also suggest

that the reported relationship is larger for US students and when panel data are used, while it

is lower when income is controlled for and in the short run. Moreover, it is remarkable that the

correlation between tuition and enrollment has been stable in time over the last 50 years.

Two qualifications of our analysis are in order. First, while we would prefer to work with

elasticities, many studies estimate the relationship between tuition fees and enrollment using

approaches other than the log-log specification. We already have to exclude a significant portion

of studies because they do not report standard errors, t-statistics, or confidence intervals for

their results, thus making it impossible for us to test the presence of publication bias. Restricting

our dataset to log-log specifications would drastically reduce the number of degrees of freedom

34

available for our analysis. While it is possible to recompute some of the other coefficients to

elasticities evaluated at the sample mean, many studies do not report the statistics necessary

for this computation. Therefore, we choose to work with partial correlation coefficients, which

can be computed easily from all the studies, and we include an analysis of the sub-sample of

elasticities only as an extension. Since our main result indicates negligible partial correlation

absent publication bias, it also directly translates into a finding of a zero mean elasticity of

demand for higher education to tuition fees. Second, the results of a meta-analysis are clearly

conditional on the quality of the previous studies included in the sample. For instance, if

all studies in the literature share a common misspecification that biases their results toward

zero, we are unable to control for such a misspecification, and our results are thus also biased.

Therefore, the correct interpretation of our analysis is that, judging from the available empirical

research, our best guess concerning the effect of tuition on enrollment is close to zero.

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39

A Supplementary Statistics and Diagnostics of BMA

Figure 5: Estimates of the tuition-enrollment nexus vary within and across studies

-1 -.5 0 .5 1

Partial correlation coefficient

Tuckman (1970)Toutkoushian & Hollis (1998)

Tannen (1978)Sulock (1982)

Shin & Milton (2008)Savoca (1990)

Quinn & Price (1998)Quigley & Rubinfeld (1993)

Paulsen & Pogue (1988)Parker & Summers (1993)

Ordovensky (1995)Noorbakhsh & Culp (2002)

Murphy & Trandel (1994)Mueller & Rockerbie (2005)

McPerson & Schapiro (1991)Koshal et al. (1976)

Knudsen & Servelle (1978)King (1993)

Kane (2007)Huijsman et al. (1986)Hsing & Chang (1996)

Hoenack & Weiler (1975)Hoenack & Pierro (1990)

Hight (1975)Hemelt & Marcotte (2011)

Grubb (1988)Elliott & Soo (2013)

Doyle & Cicarelli (1980)Dearden et al. (2011)

Craft et al. (2012)Coelli (2009)

Chressanthis (1986)Cheslock (2001)

Chen (2016)Canton & de Jong (2002)Campbell & Siegel (1967)

Buss et al. (2004)Bruckmeier et al. (2013)

Bezmen & Depken (1998)Berger & Kostal (2002)

Allen & Shen (1999)Alexander & Frey (1984)

Agarwal & Winkler (1985)

Notes: The figure shows a box plot of the partial correlation coefficients capturing the relationshipbetween tuition and the demand for higher education reported in individual studies.

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Figure 6: Estimates of the elasticity vary across different countries

-1 -.5 0 .5 1Partial correlation coefficient

USA

UK

Netherlands

Multinational

Germany

Canada

Notes: The figure shows a box plot of the partial correlation coefficients capturing the relationshipbetween tuition and the demand for higher education reported for individual countries.

Figure 7: Posterior coefficient distributions for the most important characteristics

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Cond. EV2x Cond. SDMedian

Notes: The figure depicts the densities of the regression parameters from Table 5with the highest posterior inclusion probabilities.

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Table 10: Summary of main BMA estimation

Mean no. regressors Draws Burn-ins Time No. models visited8.7563 2 · 106 1 · 105 5.495486 mins 578,591Modelspace Visited Topmodels Corr PMP No. obs.262,144 22.10% 100% 0.9988 442Model prior g-prior Shrinkage-statsUniform UIP Av = 0.9977

Notes: We employ the priors suggested by Eicher et al. (2011), who recommend using theuniform model prior (each model has the same prior probability) and the unit information prior(the prior provides the same amount of information as one observation in the data). The resultsof this BMA exercise are reported in Table 5.

Figure 8: Model size and convergence of main BMA estimation

0.00

0.10

0.20

Posterior Model Size Distribution Mean: 8.7563

Model Size

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Posterior Prior

0 1000 2000 3000 4000 5000

0.00

0.02

0.04

Posterior Model Probabilities(Corr: 0.9998)

Index of Models

PMP (MCMC) PMP (Exact)

Notes: The figure depicts the posterior model size distribution and the posterior model probabilities of theBMA exercise reported in Table 5.

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Figure 9: Correlations between the variables from Table 5

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Notes: The figure depicts the correlation coefficients between variables included in thebenchmark BMA exercise from Table 5. The definition and summary statistics of thevariables can be found in Table 4.

43

B Diagnostics of BMA robustness checks

Table 11: Summary of BMA estimation—Different BMA priors specification

Mean no. regressors Draws Burn-ins Time No. models visited8.7698 2 · 106 1 · 105 5.869949 mins 572,046Modelspace Visited Topmodels Corr PMP No. obs.262,144 21.80% 100% 0.9999 442Model prior g-prior Shrinkage-statsRandom BRIC Av = 0.9977

Notes: We employ the “random” model prior, which refers to the beta-binomial prior advocated by Ley& Steel (2009); Zellner’s g prior is set according to Fernandez et al. (2001). The results of this BMAexercise are reported in Table 7 (different BMA priors specification).

Figure 10: Model inclusion in BMA—Different BMA priors specification

0 0.09 0.19 0.26 0.34 0.42 0.49 0.57 0.64 0.72 0.79 0.86 0.94 1

Male candidates

Citations

Private universities

Panel data

OLS

Standard error

Income control

Short-run effect

USA

Publication year

Endogeneity control

Published study

Unemployment control

Cross-sectional data

Doublelog function

Female candidates

Linear function

Public universities

Notes: The figure depicts the results of the BMA related to different BMA priorsspecification reported in Table 7.

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Table 12: Summary of BMA estimation—Precision-weighted data specification

Mean no. regressors Draws Burn-ins Time No. models visited12.0906 2 · 106 1 · 105 6.009023 mins 465,235Modelspace Visited Topmodels Corr PMP No. obs.262,144 17.70% 100% 0.9995 442Model prior g-prior Shrinkage-statsUniform UIP Av = 0.9977

Notes: We employ the priors suggested by Eicher et al. (2011), who recommend using the uniform modelprior (each model has the same prior probability) and the unit information prior (the prior providesthe same amount of information as one observation in the data). The results of this BMA exercise arereported in Table 7 (precision-weighted data specification).

Figure 11: Model inclusion in BMA—Precision-weighted data specification

0 0.05 0.15 0.23 0.3 0.37 0.47 0.55 0.63 0.7 0.76 0.84 0.91 0.99

Private universities

Public universities

USA

Cross-sectional data

Linear function

Citations

Male candidates

Published study

Unemployment control

Doublelog function

Standard error

Publication year

Short-run effect

Income control

Endogeneity control

Female candidates

Panel data

OLS

Notes: The figure depicts the results of the BMA related to precision-weighted dataspecification reported in Table 7.

45